changeset 18242 2215049cd29c
parent 16417 9bc16273c2d4
child 19736 d8d0f8f51d69
--- a/src/HOL/Induct/Tree.thy	Wed Nov 23 22:26:13 2005 +0100
+++ b/src/HOL/Induct/Tree.thy	Thu Nov 24 00:00:20 2005 +0100
@@ -46,7 +46,7 @@
   "add i (Lim f) = Lim (%n. add i (f n))"
 lemma add_assoc: "add (add i j) k = add i (add j k)"
-by (induct k, auto)
+  by (induct k) auto
 text{*Multiplication of ordinals*}
@@ -57,14 +57,10 @@
   "mult i (Lim f) = Lim (%n. mult i (f n))"
 lemma add_mult_distrib: "mult i (add j k) = add (mult i j) (mult i k)"
-apply (induct k) 
-apply (auto simp add: add_assoc) 
+  by (induct k) (auto simp add: add_assoc)
 lemma mult_assoc: "mult (mult i j) k = mult i (mult j k)"
-apply (induct k) 
-apply (auto simp add: add_mult_distrib) 
+  by (induct k) (auto simp add: add_mult_distrib)
 text{*We could probably instantiate some axiomatic type classes and use
 the standard infix operators.*}
@@ -104,7 +100,6 @@
   (hints recdef_wf: wf_brouwer_order)
 lemma add2_assoc: "add2 (add2 (i, j), k) = add2 (i, add2 (j, k))"
-by (induct k, auto)
+  by (induct k) auto